Zadanie z równań różniczkowych

Interaktywne demonstracje

Należy wybrać jeden temat z pliku Mownit__ODE.pdf

Rozwiązaniem jest program pokazujący w interaktywny sposób rozwiązanie danego problemu obliczeniowego. Program powinien mieć graficzny interfejs użytkownika oraz umozliwiać dynamiczną interakcję poprzez zmianę parametrów. Język programowania oraz środowisko: dowolne. Proponowane rozwiązania:

• Dash: https://dash.plotly.com/
• Pakiet Shiny w R: https://shiny.rstudio.com
• Julia + AngularJS + Plotly, przykład: https://github.com/JuliaDiffEq/DiffEqOnline
• Julia interact, przykład: https://github.com/JuliaGizmos/Interact.jl
• C + GSL + JavaFX ... Jako rozwiązanie należy załączyć plik Jupyter notebook lub HTML zawierający np. link do nagranego filmiku z demonstracją działania programu.

Zadanie 2-tygodniowe

SIR epidemic model

Model ten dzieli rozpatrywaną populację na trzy stany:

• S (susceptible) - osoby, które jeszcze nie zostały zainfekowane, nie mają odporności, są więc podatne na zarażenie
• I (infected) - osoby obecnie zainfekowane ("chore") oraz zaraźliwe dla sąsiadów w stanie S
• R (removed) - niebrane pod uwagę w dalszym procesie z powodu uzyskania stałej odporności lub z powodu śmierci

Liczba osób w danym stanie opisywana jest w funkcji czasu, odpowiednio I, S albo R. Jednocześnie w danym czasie t suma wartości tych trzech funkcji w tym punkcie czasu wynosi N (liczebność rozpatrywanej populacji). Model SIR bazuje na następujących równaniach różniczkowych, opisujących tempo przyrostu liczebności poszczególnych grup:

gdzie:

• $\beta$ - współczynnik przenoszenia choroby: opisuje prawodpodobieństwo transmisji choroby z osoby z grupy I na osobę z grupy R
• $\gamma$ - współczynnik powrotu do zdrowia: opisuje tempo przejścia osób z grupy I do grupy R. Jest odwrotnie proporcjonalny do czasu potrzebnego do wyzdrowienia z choroby, wyrażonego w dniach, n

Współczynnik reprodukcji

Stanowi ważny parametr opisu dynamiki epidemii. Określa on średnią liczbę osób zakażonych przez jedną zainfekowaną osobę. $$R_{0}=\frac{\beta}{\gamma}$$

Osoby, które przebyły chorobę COVID-19, wykształcają odporność trwającą przez co najmniej 5-7 miesięcy. Stanowi to na tyle długi okres czasu, że tzw. "ozdrowieńców" zakwalifikować można do grupy R, co umożliwia zastosowanie modelu SIR w badaniu przebiegu epidemii SARS-COV-2.

Wykresy zmian liczby osób w poszczególnych stanach

using DifferentialEquations
# using Plotly
 using Plots
# plotly()
# Plots.PlotlyBackend()
# using PlotlyBase
# using PyPlot

• Obliczanie pochodnych w punktach czasu

function parameterized_derivative!(du, u, p, t)
    S, I, R = u
    beta, gamma, N = p
    du[1] = -beta * S * I / N
    du[2] = beta * S * I / N - gamma * I
    du[3] = gamma * I
end

parameterized_derivative! (generic function with 1 method)

• Rozwiązywanie układu równań różniczkowych dla danych parametrów modelu SIR

function SIR_computations(parameters, variables, days)
    beta = parameters[1]
    gamma = parameters[2]
    N = parameters[3]
    
    I0 = variables[1]
    R0 = variables[2]
    S0 = variables[3]
    
    # Define differential equation problem
    u0 = [S0; I0; R0]
    tspan = (0.0, days)
    params = [ beta, gamma, N]
    prob = ODEProblem(parameterized_derivative!, u0, tspan, params);
    alg = RK4()

    # Solve system of differential equations
    sol = solve(prob, alg);
    return sol
end

SIR_computations (generic function with 1 method)

Wykres dla $\beta$ = 0.15, $\gamma$ = 1/21

# Define model parameters
N = 1000.0
beta = 0.15
gamma = 1.0/21.0

# Define population
I0 = 1;
R0 = 0;
S0 = N - I0 - R0;

parameters = (beta, gamma, N)
variables = (I0, R0, S0)
days = 200.0

# Perform computations
result = SIR_computations(parameters, variables, days)

# Plot results
plot1 = Plots.plot(result, vars=[(0,1), (0,2), (0,3)], title="SIR Model:\n beta = 0.15, gamma = 1/21", 
    xaxis="Time [days]", yaxis="Population", marker=0.01, label=["Susceptible (S)" "Infected (I)" "Removed (R)"],
    xticks = 0:25:200, yticks = 0:100:1000)

Wykres dla $\beta$ = 0.15, $\gamma$ = 1/14

# Define model parameters
N = 1000.0
beta = 0.15
gamma = 1.0/14.0

# Define population
I0 = 1;
R0 = 0;
S0 = N - I0 - R0;

parameters = (beta, gamma, N)
variables = (I0, R0, S0)
days = 200.0

# Perform computations
result = SIR_computations(parameters, variables, days);

# Plot results
plot2 = Plots.plot(result, vars=[(0,1), (0,2), (0,3)], title="SIR Model:\n beta = 0.15, gamma = 1/14", 
    xaxis="Time [days]", yaxis="Population", marker=0.01, label=["Susceptible (S)" "Infected (I)" "Removed (R)"],
    xticks = 0:25:200, yticks = 0:100:1000)

Porównanie wykresów dla różnego czasu zdrowienia n = 1/$\gamma$

Plots.plot(plot1, plot2, layout=(1,2), size=(1100,500))

Zestawienie powyższych wykresów ilustruje zależność dynamiki przebiegu epidemii od czasu zdrowienia. Przyglądając się punktowi tzw. odporności zbiorowej, przyjmowanej jako R=70%, widać, że jest jest ona osiągana szybiej, gdy czas zdrowienia jest dłuższy (21 dni w porównaniu do 14 dni). Dzieje się tak ponieważ osobniki zakażone mają szansę zarażać dłużej, zanim wyzdrowieją, zatem fala zakażeń ma bardziej dynamiczny przebieg, większa liczba osób szybciej przechoruje, w konsekwencji czego odporność zbiorowa zostanie uzyskana szybciej.

Część Interaktywna

# using IJulia
# using Conda
# using Interact
# using WebIO
# WebIO.install_jupyter_labextension()
#notebook(dir = Interact.notebookdir)
# using WebIO
# WebIO.install_jupyter_labextension(condajl=true)

using Mux
#WebIO.webio_serve(page("/", req -> ui), rand(8000:9000))
WebIO.webio_serve(page("/", req -> ui), 9000)

Task (runnable) @0x000000008367ea40

using Interact
using Blink
ui = button("Hi")
#display(ui)
w = Window()
o = observe(ui)
o[]
on(println, o)
on(n -> println(3+3), o)
o[] = 33;
t = textbox() 
body!(w, ui);

33
6
34
6
34
6

# Define population
I0 = 1;
R0 = 0;

days = 200.0

import Colors
using Plots

function mycolorpicker()
    N = slider(1:1000, label = "N")
    beta = slider(0:0.01:1, label = "beta")
    gamma = slider(0:0.01:1, label = "gamma")
    parameters = Interact.@map (&beta, &gamma, &N)
    S0 = Interact.@map &N - I0 - R0;
    variables = Interact.@map (I0, R0, &S0)
    output = Interact.@map SIR_computations(&parameters, &variables, days);
    plt = Interact.@map plot(&output, vars=[(0,1), (0,2), (0,3)], title="SIR Model:\n beta = 0.15, gamma = 1/14", 
    xaxis="Time [days]", yaxis="Population", marker=0.01, label=["Susceptible (S)" "Infected (I)" "Removed (R)"],
    xticks = 0:25:200, yticks = 0:100:1000)
    wdg = Widget(["N" => N, "beta" => beta, "gamma" => gamma], output = output)
    @layout! wdg hbox(plt, vbox(:N, :beta, :gamma), plt) ## custom layout: by default things are stacked vertically
end

mycolorpicker (generic function with 1 method)

ui = mycolorpicker()
o=Observable(ui)
ui

#using Blink
#w = Window()
#body!(w, ui)

Error handling websocket connection:
TaskFailedException
Stacktrace:
  [1] wait
    @ .\task.jl:317 [inlined]
  [2] create_socket(req::Dict{Any, Any})
    @ WebIO ~\.julia\packages\WebIO\Fy9h1\src\providers\mux.jl:47
  [3] (::Mux.var"#5#6"{Mux.var"#31#32"{Vector{SubString{String}}}, typeof(WebIO.create_socket)})(f::Function, x::Dict{Any, Any})
    @ Mux ~\.julia\packages\Mux\3h8RY\src\Mux.jl:17
  [4] (::Mux.var"#1#2"{Mux.var"#5#6"{Mux.var"#31#32"{Vector{SubString{String}}}, typeof(WebIO.create_socket)}, Mux.var"#1#2"{typeof(Mux.wclose), Mux.var"#1#2"{Mux.var"#21#22"{Mux.var"#25#26"{Symbol, Int64}}, Mux.var"#23#24"{String}}}})(x::Dict{Any, Any})
    @ Mux ~\.julia\packages\Mux\3h8RY\src\Mux.jl:10
  [5] splitquery(app::Mux.var"#1#2"{Mux.var"#5#6"{Mux.var"#31#32"{Vector{SubString{String}}}, typeof(WebIO.create_socket)}, Mux.var"#1#2"{typeof(Mux.wclose), Mux.var"#1#2"{Mux.var"#21#22"{Mux.var"#25#26"{Symbol, Int64}}, Mux.var"#23#24"{String}}}}, req::Dict{Any, Any})
    @ Mux ~\.julia\packages\Mux\3h8RY\src\basics.jl:31
  [6] #1
    @ ~\.julia\packages\Mux\3h8RY\src\Mux.jl:10 [inlined]
  [7] wcatch(app::Mux.var"#1#2"{typeof(Mux.splitquery), Mux.var"#1#2"{Mux.var"#5#6"{Mux.var"#31#32"{Vector{SubString{String}}}, typeof(WebIO.create_socket)}, Mux.var"#1#2"{typeof(Mux.wclose), Mux.var"#1#2"{Mux.var"#21#22"{Mux.var"#25#26"{Symbol, Int64}}, Mux.var"#23#24"{String}}}}}, req::Dict{Any, Any})
    @ Mux ~\.julia\packages\Mux\3h8RY\src\websockets_integration.jl:12
  [8] #1
    @ ~\.julia\packages\Mux\3h8RY\src\Mux.jl:10 [inlined]
  [9] todict
    @ ~\.julia\packages\Mux\3h8RY\src\basics.jl:25 [inlined]
 [10] #3 (repeats 2 times)
    @ ~\.julia\packages\Mux\3h8RY\src\Mux.jl:14 [inlined]
 [11] (::Mux.var"#1#2"{Mux.var"#3#4"{Mux.var"#3#4"{typeof(Mux.todict), typeof(Mux.wcatch)}, typeof(Mux.splitquery)}, Mux.var"#1#2"{Mux.var"#5#6"{Mux.var"#31#32"{Vector{SubString{String}}}, typeof(WebIO.create_socket)}, Mux.var"#1#2"{typeof(Mux.wclose), Mux.var"#1#2"{Mux.var"#21#22"{Mux.var"#25#26"{Symbol, Int64}}, Mux.var"#23#24"{String}}}}})(x::Tuple{HTTP.Messages.Request, WebSockets.WebSocket{Sockets.TCPSocket}})
    @ Mux ~\.julia\packages\Mux\3h8RY\src\Mux.jl:10
 [12] (::Mux.var"#9#10"{Mux.App})(req::HTTP.Messages.Request, client::WebSockets.WebSocket{Sockets.TCPSocket})
    @ Mux ~\.julia\packages\Mux\3h8RY\src\server.jl:49
 [13] upgrade(f::Mux.var"#9#10"{Mux.App}, stream::HTTP.Streams.Stream{HTTP.Messages.Request, HTTP.ConnectionPool.Transaction{Sockets.TCPSocket}})
    @ WebSockets ~\.julia\packages\WebSockets\QcswW\src\HTTP.jl:201
 [14] (::WebSockets.var"#_servercoroutine#11"{WebSockets.ServerWS})(stream::HTTP.Streams.Stream{HTTP.Messages.Request, HTTP.ConnectionPool.Transaction{Sockets.TCPSocket}})
    @ WebSockets ~\.julia\packages\WebSockets\QcswW\src\HTTP.jl:370
 [15] macro expansion
    @ ~\.julia\packages\HTTP\IAI92\src\Servers.jl:367 [inlined]
 [16] (::HTTP.Servers.var"#13#14"{WebSockets.var"#_servercoroutine#11"{WebSockets.ServerWS}, HTTP.ConnectionPool.Transaction{Sockets.TCPSocket}, HTTP.Streams.Stream{HTTP.Messages.Request, HTTP.ConnectionPool.Transaction{Sockets.TCPSocket}}})()
    @ HTTP.Servers .\task.jl:406

    nested task error: WebSockets.WebSocketClosedError("ws|server respond to OPCODE_CLOSE 1001: Going Away")
    Stacktrace:
     [1] handle_control_frame(ws::WebSockets.WebSocket{Sockets.TCPSocket}, wsf::WebSockets.WebSocketFragment)
       @ WebSockets ~\.julia\packages\WebSockets\QcswW\src\WebSockets.jl:376
     [2] read(ws::WebSockets.WebSocket{Sockets.TCPSocket})
       @ WebSockets ~\.julia\packages\WebSockets\QcswW\src\WebSockets.jl:459
     [3] macro expansion
       @ ~\.julia\packages\WebIO\Fy9h1\src\providers\mux.jl:41 [inlined]
     [4] (::WebIO.var"#110#111"{WebIO.WebSockConnection, WebSockets.WebSocket{Sockets.TCPSocket}})()
       @ WebIO .\task.jl:406┌ Error: (Base.IOError("read: connection reset by peer (ECONNRESET)", -4077), Base.StackTraces.StackFrame[wait_readnb(x::Sockets.TCPSocket, nb::Int64) at stream.jl:394, eof at stream.jl:106 [inlined], read_to_buffer(t::HTTP.ConnectionPool.Transaction{Sockets.TCPSocket}, sizehint::Int64) at ConnectionPool.jl:251, readuntil(t::HTTP.ConnectionPool.Transaction{Sockets.TCPSocket}, f::Function, sizehint::Int64) at ConnectionPool.jl:271, readuntil at ConnectionPool.jl:269 [inlined], readheaders at Messages.jl:471 [inlined], startread(http::HTTP.Streams.Stream{HTTP.Messages.Request, HTTP.ConnectionPool.Transaction{Sockets.TCPSocket}}) at Streams.jl:155, handle_transaction(f::WebSockets.var"#_servercoroutine#11"{WebSockets.ServerWS}, t::HTTP.ConnectionPool.Transaction{Sockets.TCPSocket}, server::HTTP.Servers.Server{Nothing, Sockets.TCPServer}; final_transaction::Bool) at Servers.jl:343, (::HTTP.Servers.var"#handle_transaction##kw")(::NamedTuple{(:final_transaction,), Tuple{Bool}}, ::typeof(HTTP.Servers.handle_transaction), f::Function, t::HTTP.ConnectionPool.Transaction{Sockets.TCPSocket}, server::HTTP.Servers.Server{Nothing, Sockets.TCPServer}) at Servers.jl:338, handle_connection(f::Function, c::HTTP.ConnectionPool.Connection{Sockets.TCPSocket}, server::HTTP.Servers.Server{Nothing, Sockets.TCPServer}, reuse_limit::Int64, readtimeout::Int64) at Servers.jl:299, (::HTTP.Servers.var"#8#9"{WebSockets.var"#_servercoroutine#11"{WebSockets.ServerWS}, HTTP.Servers.Server{Nothing, Sockets.TCPServer}, Base.RefValue{Int64}, Int64, Int64, Bool, HTTP.ConnectionPool.Connection{Sockets.TCPSocket}})() at task.jl:406])
└ @ HTTP.Servers C:\Users\basia\.julia\packages\HTTP\IAI92\src\Servers.jl:264

# Define model parameters
beta = 0.15
gamma = 1.0/14.0

# Initial conditions (populations)
I0 = 1;
R0 = 0;


# Define additional parameters
days = 200
h = 1

# Define the ODE functions
dS(S, I, R, N, beta, gamma) = -beta * S * I / N
dI(S, I, R, N, beta, gamma) = beta * S * I / N - gamma * I
dR(S, I, R, N, beta, gamma) = gamma * I

funcs = (dS, dI, dR)

using Interact
using Plots
#=
using InteractBulma

timer = Observable(0.0)
@async while false
    sleep(0.1)
    timer[] = timer[]+0.1
end
=#

Ns = slider(1:1000; label = "N")
Betas = slider(0:0.01:1.0; label = "Beta")
Gammas = slider(0:0.01:1.0; label = "Gamma")

function create_params(beta, gamma, N, days, h)
    return (beta, gamma, N, days, h)
end

function create_init_conds(S0, I0, R0)
    return (S0, I0, R0)
end

function get_S0(N, I0, R0)
    return N - I0 - R0
end

ui2 = @manipulate for N in Ns, Beta in Betas, Gamma in Gammas
    parameters = create_params(Beta, Gamma, N, days, h)
    S0 = get_S0(N, I0, R0)
    init_conds = (S0, I0, R0)
    output = Runge_Kutta(init_conds, parameters, funcs)
    plt = plot([output[1] output[2] output[3]], vars=[(0,1), (0,2), (0,3)], title="SIR Model:\n beta = 0.15, gamma = 1/14", 
    xaxis="Time [days]", yaxis="Population", marker=0.01, label=["Susceptible (S)" "Infected (I)" "Removed (R)"],
    xticks = 0:25:200, yticks = 0:100:1000)
    wdg = Widget(["N" => N, "Beta" => Beta, "Gamma" => Gamma])
    @layout! wdg hbox(plt, vbox(:N, :Beta, :Gamma)) ## custom layout: by default things are stacked vertically
end
ui2


#=
import Colors
using Plots

function mycolorpicker()
    N = slider(1:1000, label = "N")
    beta = slider(0:0.01:1, label = "beta")
    gamma = slider(0:0.01:1, label = "gamma")
    parameters = Interact.@map (&beta, &gamma, &N)
    S0 = Interact.@map &N - I0 - R0;
    variables = Interact.@map (I0, R0, &S0)
    output = Interact.@map SIR_computations(&parameters, &variables, days);
    plt = Interact.@map plot(&output, vars=[(0,1), (0,2), (0,3)], title="SIR Model:\n beta = 0.15, gamma = 1/14", 
    xaxis="Time [days]", yaxis="Population", marker=0.01, label=["Susceptible (S)" "Infected (I)" "Removed (R)"],
    xticks = 0:25:200, yticks = 0:100:1000)
    wdg = Widget(["N" => N, "beta" => beta, "gamma" => gamma], output = output)
    @layout! wdg hbox(plt, vbox(:N, :beta, :gamma), plt) ## custom layout: by default things are stacked vertically
end
=#

using Plots

timer = Observable(0.0)
@async while true
    sleep(0.1)
    timer[] = timer[]+0.1
end

x = y = 0:0.1:30

freqs = OrderedDict(zip(["pi/4", "π/2", "3π/4", "π"], [π/4, π/2, 3π/4, π]))

ui2 = @manipulate for freq1 in freqs, freq2 in slider(0.01:0.1:4π; label="freq2"), t=timer
    y = @. [sin(freq1*x.+t) sin(freq1*x)]
    plot(x, y)
end
ui2

width, height = 700, 300
colors = ["black", "gray", "silver", "maroon", "red", "olive", "yellow", "green", "lime", "teal", "aqua", "navy", "blue", "purple", "fuchsia"]
color(i) = colors[i%length(colors)+1]
ui = @manipulate for nsamples in 1:200,
        sample_step in slider(0.01:0.01:1.0, value=0.1, label="sample step"),
        phase in slider(0:0.1:2pi, value=0.0, label="phase"),
        radii in 0.1:0.1:60
    cxs_unscaled = [i*sample_step + phase for i in 1:nsamples]
    cys = sin.(cxs_unscaled) .* height/3 .+ height/2
    cxs = cxs_unscaled .* width/4pi
    dom"svg:svg[width=$width, height=$height]"(
        (dom"svg:circle[cx=$(cxs[i]), cy=$(cys[i]), r=$radii, fill=$(color(i))]"()
            for i in 1:nsamples)...
    )
end

Error handling websocket connection:
TaskFailedException
Stacktrace:
  [1] wait
    @ .\task.jl:317 [inlined]
  [2] create_socket(req::Dict{Any, Any})
    @ WebIO ~\.julia\packages\WebIO\Fy9h1\src\providers\mux.jl:47
  [3] (::Mux.var"#5#6"{Mux.var"#31#32"{Vector{SubString{String}}}, typeof(WebIO.create_socket)})(f::Function, x::Dict{Any, Any})
    @ Mux ~\.julia\packages\Mux\3h8RY\src\┌ Error: error handling request
│   exception = (Base.IOError("stream is closed or unusable", 0), Base.StackTraces.StackFrame[check_open at stream.jl:386 [inlined], uv_write_async(s::Sockets.TCPSocket, p::Ptr{UInt8}, n::UInt64) at stream.jl:1018, uv_write(s::Sockets.TCPSocket, p::Ptr{UInt8}, n::UInt64) at stream.jl:981, unsafe_write(s::Sockets.TCPSocket, p::Ptr{UInt8}, n::UInt64) at stream.jl:1064, unsafe_write at ConnectionPool.jl:171 [inlined], write at io.jl:185 [inlined], closebody at Streams.jl:113 [inlined], closewrite(http::HTTP.Streams.Stream{HTTP.Messages.Request, HTTP.ConnectionPool.Transaction{Sockets.TCPSocket}}) at Streams.jl:128, (::HTTP.Servers.var"#13#14"{WebSockets.var"#_servercoroutine#11"{WebSockets.ServerWS}, HTTP.ConnectionPool.Transaction{Sockets.TCPSocket}, HTTP.Streams.Stream{HTTP.Messages.Request, HTTP.ConnectionPool.Transaction{Sockets.TCPSocket}}})() at task.jl:406])
└ @ HTTP.Servers C:\Users\basia\.julia\packages\HTTP\IAI92\src\Servers.jl:373
Mux.jl:17
  [4] (::Mux.var"#1#2"{Mux.var"#5#6"{Mux.var"#31#32"{Vector{SubString{String}}}, typeof(WebIO.create_socket)}, Mux.var"#1#2"{typeof(Mux.wclose), Mux.var"#1#2"{Mux.var"#21#22"{Mux.var"#25#26"{Symbol, Int64}}, Mux.var"#23#24"{String}}}})(x::Dict{Any, Any})
    @ Mux ~\.julia\packages\Mux\3h8RY\src\Mux.jl:10
  [5] splitquery(app::Mux.var"#1#2"{Mux.var"#5#6"{Mux.var"#31#32"{Vector{SubString{String}}}, typeof(WebIO.create_socket)}, Mux.var"#1#2"{typeof(Mux.wclose), Mux.var"#1#2"{Mux.var"#21#22"{Mux.var"#25#26"{Symbol, Int64}}, Mux.var"#23#24"{String}}}}, req::Dict{Any, Any})
    @ Mux ~\.julia\packages\Mux\3h8RY\src\basics.jl:31
  [6] #1
    @ ~\.julia\packages\Mux\3h8RY\src\Mux.jl:10 [inlined]
  [7] wcatch(app::Mux.var"#1#2"{typeof(Mux.splitquery), Mux.var"#1#2"{Mux.var"#5#6"{Mux.var"#31#32"{Vector{SubString{String}}}, typeof(WebIO.create_socket)}, Mux.var"#1#2"{typeof(Mux.wclose), Mux.var"#1#2"{Mux.var"#21#22"{Mux.var"#25#26"{Symbol, Int64}}, Mux.var"#23#24"{String}}}}}, req::Dict{Any, Any})
    @ Mux ~\.julia\packages\Mux\3h8RY\src\websockets_integration.jl:12
  [8] #1
    @ ~\.julia\packages\Mux\3h8RY\src\Mux.jl:10 [inlined]
  [9] todict
    @ ~\.julia\packages\Mux\3h8RY\src\basics.jl:25 [inlined]
 [10] #3 (repeats 2 times)
    @ ~\.julia\packages\Mux\3h8RY\src\Mux.jl:14 [inlined]
 [11] (::Mux.var"#1#2"{Mux.var"#3#4"{Mux.var"#3#4"{typeof(Mux.todict), typeof(Mux.wcatch)}, typeof(Mux.splitquery)}, Mux.var"#1#2"{Mux.var"#5#6"{Mux.var"#31#32"{Vector{SubString{String}}}, typeof(WebIO.create_socket)}, Mux.var"#1#2"{typeof(Mux.wclose), Mux.var"#1#2"{Mux.var"#21#22"{Mux.var"#25#26"{Symbol, Int64}}, Mux.var"#23#24"{String}}}}})(x::Tuple{HTTP.Messages.Request, WebSockets.WebSocket{Sockets.TCPSocket}})
    @ Mux ~\.julia\packages\Mux\3h8RY\src\Mux.jl:10
 [12] (::Mux.var"#9#10"{Mux.App})(req::HTTP.Messages.Request, client::WebSockets.WebSocket{Sockets.TCPSocket})
    @ Mux ~\.julia\packages\Mux\3h8RY\src\server.jl:49
 [13] upgrade(f::Mux.var"#9#10"{Mux.App}, stream::HTTP.Streams.Stream{HTTP.Messages.Request, HTTP.ConnectionPool.Transaction{Sockets.TCPSocket}})
    @ WebSockets ~\.julia\packages\WebSockets\QcswW\src\HTTP.jl:201
 [14] (::WebSockets.var"#_servercoroutine#11"{WebSockets.ServerWS})(stream::HTTP.Streams.Stream{HTTP.Messages.Request, HTTP.ConnectionPool.Transaction{Sockets.TCPSocket}})
    @ WebSockets ~\.julia\packages\WebSockets\QcswW\src\HTTP.jl:370
 [15] macro expansion
    @ ~\.julia\packages\HTTP\IAI92\src\Servers.jl:367 [inlined]
 [16] (::HTTP.Servers.var"#13#14"{WebSockets.var"#_servercoroutine#11"{WebSockets.ServerWS}, HTTP.ConnectionPool.Transaction{Sockets.TCPSocket}, HTTP.Streams.Stream{HTTP.Messages.Request, HTTP.ConnectionPool.Transaction{Sockets.TCPSocket}}})()
    @ HTTP.Servers .\task.jl:406

    nested task error: WebSockets.WebSocketClosedError("ws|server respond to OPCODE_CLOSE 1001: Going Away")
    Stacktrace:
     [1] handle_control_frame(ws::WebSockets.WebSocket{Sockets.TCPSocket}, wsf::WebSockets.WebSocketFragment)
       @ WebSockets ~\.julia\packages\WebSockets\QcswW\src\WebSockets.jl:376
     [2] read(ws::WebSockets.WebSocket{Sockets.TCPSocket})
       @ WebSockets ~\.julia\packages\WebSockets\QcswW\src\WebSockets.jl:459
     [3] macro expansion
       @ ~\.julia\packages\WebIO\Fy9h1\src\providers\mux.jl:41 [inlined]
     [4] (::WebIO.var"#110#111"{WebIO.WebSockConnection, WebSockets.WebSocket{Sockets.TCPSocket}})()
       @ WebIO .\task.jl:406

35
6

Error handling websocket connection:
TaskFailedException
Stacktrace:
  [1] wait
    @ .\task.jl:317 [inlined]
  [2] create_socket(req::Dict{Any, Any})
    @ WebIO ~\.julia\packages\WebIO\Fy9h1\src\providers\mux.jl:47
  [3] (::Mux.var"#5#6"{Mux.var"#31#32"{Vector{SubString{String}}}, typeof(WebIO.create_socket)})(f::Function, x::Dict{Any, Any})
    @ Mux ~\.julia\packages\Mux\3h8RY\src\Mux.jl:17
  [4] (::Mux.var"#1#2"{Mux.var"#5#6"{Mux.var"#31#32"{Vector{SubString{String}}}, typeof(WebIO.create_socket)}, Mux.var"#1#2"{typeof(Mux.wclose), Mux.var"#1#2"{Mux.var"#21#22"{Mux.var"#25#26"{Symbol, Int64}}, Mux.var"#23#24"{String}}}})(x::Dict{Any, Any})
    @ Mux ~\.julia\packages\Mux\3h8RY\src\Mux.jl:10
  [5] splitquery(app::Mux.var"#1#2"{Mux.var"#5#6"{Mux.var"#31#32"{Vector{SubString{String}}}, typeof(WebIO.create_socket)}, Mux.var"#1#2"{typeof(Mux.wclose), Mux.var"#1#2"{Mux.var"#21#22"{Mux.var"#25#26"{Symbol, Int64}}, Mux.var"#23#24"{String}}}}, req::Dict{Any, Any})
    @ Mux ~\.julia\packages\Mux\3h8RY\src\basics.jl:31
  [6] #1
    @ ~\.julia\packages\Mux\3h8RY\src\Mux.jl:10 [inlined]
  [7] wcatch(app::Mux.var"#1#2"{typeof(Mux.splitquery), Mux.var"#1#2"{Mux.var"#5#6"{Mux.var"#31#32"{Vector{SubString{String}}}, typeof(WebIO.create_socket)}, Mux.var"#1#2"{typeof(Mux.wclose), Mux.var"#1#2"{Mux.var"#21#22"{Mux.var"#25#26"{Symbol, Int64}}, Mux.var"#23#24"{String}}}}}, req::Dict{Any, Any})
    @ Mux ~\.julia\packages\Mux\3h8RY\src\websockets_integration.jl:12
  [8] #1
    @ ~\.julia\packages\Mux\3h8RY\src\Mux.jl:10 [inlined]
  [9] todict
    @ ~\.julia\packages\Mux\3h8RY\src\basics.jl:25 [inlined]
 [10] #3 (repeats 2 times)
    @ ~\.julia\packages\Mux\3h8RY\src\Mux.jl:14 [inlined]
 [11] (::Mux.var"#1#2"{Mux.var"#3#4"{Mux.var"#3#4"{typeof(Mux.todict), typeof(Mux.wcatch)}, typeof(Mux.splitquery)}, Mux.var"#1#2"{Mux.var"#5#6"{Mux.var"#31#32"{Vector{SubString{String}}}, typeof(WebIO.create_socket)}, Mux.var"#1#2"{typeof(Mux.wclose), Mux.var"#1#2"{Mux.var"#21#22"{Mux.var"#25#26"{Symbol, Int64}}, Mux.var"#23#24"{String}}}}})(x::Tuple{HTTP.Messages.Request, WebSockets.WebSocket{Sockets.TCPSocket}})
    @ Mux ~\.julia\packages\Mux\3h8RY\src\Mux.jl:10
 [12] (::Mux.var"#9#10"{Mux.App})(req::HTTP.Messages.Request, client::WebSockets.WebSocket{Sockets.TCPSocket})
    @ Mux ~\.julia\packages\Mux\3h8RY\src\server.jl:49
 [13] upgrade(f::Mux.var"#9#10"{Mux.App}, stream::HTTP.Streams.Stream{HTTP.Messages.Request, HTTP.ConnectionPool.Transaction{Sockets.TCPSocket}})
    @ WebSockets ~\.julia\packages\WebSockets\QcswW\src\HTTP.jl:201
 [14] (::WebSockets.var"#_servercoroutine#11"{WebSockets.ServerWS})(stream::HTTP.Streams.Stream{HTTP.Messages.Request, HTTP.ConnectionPool.Transaction{Sockets.TCPSocket}})
    @ WebSockets ~\.julia\packages\WebSockets\QcswW\src\HTTP.jl:370
 [15] macro expansion
    @ ~\.julia\packages\HTTP\IAI92\src\Servers.jl:367 [inlined]
 [16] (::HTTP.Servers.var"#13#14"{WebSockets.var"#_servercoroutine#11"{WebSockets.ServerWS}, HTTP.ConnectionPool.Transaction{Sockets.TCPSocket}, HTTP.Streams.Stream{HTTP.Messages.Request, HTTP.ConnectionPool.Transaction{Sockets.TCPSocket}}})()
    @ HTTP.Servers .\task.jl:406

    nested task error: WebSockets.WebSocketClosedError("ws|server respond to OPCODE_CLOSE 1001: Going Away")
    Stacktrace:
     [1] handle_control_frame(ws::WebSockets.WebSocket{Sockets.TCPSocket}, wsf::WebSockets.WebSocketFragment)
       @ WebSockets ~\.julia\packages\WebSockets\QcswW\src\WebSockets.jl:376
     [2] read(ws::WebSockets.WebSocket{Sockets.TCPSocket})
       @ WebSockets ~\.julia\packages\WebSockets\QcswW\src\WebSockets.jl:459
     [3] macro expansion
       @ ~\.julia\packages\WebIO\Fy9h1\src\providers\mux.jl:41 [inlined]
     [4] (::WebIO.var"#110#111"{WebIO.WebSockConnection, WebSockets.WebSocket{Sockets.TCPSocket}})()
       @ WebIO .\task.jl:406

loadbutton = filepicker()
hellobutton = button("Hello!")
goodbyebutton = button("Good bye!")
b = vbox( # put things one on top of the other
    loadbutton,
    hbox( # put things one next to the other
        pad(1em, hellobutton), # to allow some white space around the widget
        pad(1em, goodbyebutton),
    )
)
display(ui)

Error handling websocket connection:
TaskFailedException
Stacktrace:
  [1] wait
    @ .\task.jl:317 [inlined]
  [2] create_socket(req::Dict{Any, Any})
    @ WebIO ~\.julia\packages\WebIO\Fy9h1\src\providers\mux.jl:47
  [3] (::Mux.var"#5#6"{Mux.var"#31#32"{Vector{SubString{String}}}, typeof(WebIO.create_socket)})(f::Function, x::Dict{Any, Any})
    @ Mux ~\.julia\packages\Mux\3h8RY\src\Mux.jl:17
  [4] (::Mux.var"#1#2"{Mux.var"#5#6"{Mux.var"#31#32"{Vector{SubString{String}}}, typeof(WebIO.create_socket)}, Mux.var"#1#2"{typeof(Mux.wclose), Mux.var"#1#2"{Mux.var"#21#22"{Mux.var"#25#26"{Symbol, Int64}}, Mux.var"#23#24"{String}}}})(x::Dict{Any, Any})
    @ Mux ~\.julia\packages\Mux\3h8RY\src\Mux.jl:10
  [5] splitquery(app::Mux.var"#1#2"{Mux.var"#5#6"{Mux.var"#31#32"{Vector{SubString{String}}}, typeof(WebIO.create_socket)}, Mux.var"#1#2"{typeof(Mux.wclose), Mux.var"#1#2"{Mux.var"#21#22"{Mux.var"#25#26"{Symbol, Int64}}, Mux.var"#23#24"{String}}}}, req::Dict{Any, Any})
    @ Mux ~\.julia\packages\Mux\3h8RY\src\basics.jl:31
  [6] #1
    @ ~\.julia\packages\Mux\3h8RY\src\Mux.jl:10 [inlined]
  [7] wcatch(app::Mux.var"#1#2"{typeof(Mux.splitquery), Mux.var"#1#2"{Mux.var"#5#6"{Mux.var"#31#32"{Vector{SubString{String}}}, typeof(WebIO.create_socket)}, Mux.var"#1#2"{typeof(Mux.wclose), Mux.var"#1#2"{Mux.var"#21#22"{Mux.var"#25#26"{Symbol, Int64}}, Mux.var"#23#24"{String}}}}}, req::Dict{Any, Any})
    @ Mux ~\.julia\packages\Mux\3h8RY\src\websockets_integration.jl:12
  [8] #1
    @ ~\.julia\packages\Mux\3h8RY\src\Mux.jl:10 [inlined]
  [9] todict
    @ ~\.julia\packages\Mux\3h8RY\src\basics.jl:25 [inlined]
 [10] #3 (repeats 2 times)
    @ ~\.julia\packages\Mux\3h8RY\src\Mux.jl:14 [inlined]
 [11] (::Mux.var"#1#2"{Mux.var"#3#4"{Mux.var"#3#4"{typeof(Mux.todict), typeof(Mux.wcatch)}, typeof(Mux.splitquery)}, Mux.var"#1#2"{Mux.var"#5#6"{Mux.var"#31#32"{Vector{SubString{String}}}, typeof(WebIO.create_socket)}, Mux.var"#1#2"{typeof(Mux.wclose), Mux.var"#1#2"{Mux.var"#21#22"{Mux.var"#25#26"{Symbol, Int64}}, Mux.var"#23#24"{String}}}}})(x::Tuple{HTTP.Messages.Request, WebSockets.WebSocket{Sockets.TCPSocket}})
    @ Mux ~\.julia\packages\Mux\3h8RY\src\Mux.jl:10
 [12] (::Mux.var"#9#10"{Mux.App})(req::HTTP.Messages.Request, client::WebSockets.WebSocket{Sockets.TCPSocket})
    @ Mux ~\.julia\packages\Mux\3h8RY\src\server.jl:49
 [13] upgrade(f::Mux.var"#9#10"{Mux.App}, stream::HTTP.Streams.Stream{HTTP.Messages.Request, HTTP.ConnectionPool.Transaction{Sockets.TCPSocket}})
    @ WebSockets ~\.julia\packages\WebSockets\QcswW\src\HTTP.jl:201
 [14] (::WebSockets.var"#_servercoroutine#11"{WebSockets.ServerWS})(stream::HTTP.Streams.Stream{HTTP.Messages.Request, HTTP.ConnectionPool.Transaction{Sockets.TCPSocket}})
    @ WebSockets ~\.julia\packages\WebSockets\QcswW\src\HTTP.jl:370
 [15] macro expansion
    @ ~\.julia\packages\HTTP\IAI92\src\Servers.jl:367 [inlined]
 [16] (::HTTP.Servers.var"#13#14"{WebSockets.var"#_servercoroutine#11"{WebSockets.ServerWS}, HTTP.ConnectionPool.Transaction{Sockets.TCPSocket}, HTTP.Streams.Stream{HTTP.Messages.Request, HTTP.ConnectionPool.Transaction{Sockets.TCPSocket}}})()
    @ HTTP.Servers .\task.jl:406

    nested task error: WebSockets.WebSocketClosedError("ws|server respond to OPCODE_CLOSE 1001: Going Away")
    Stacktrace:
     [1] handle_control_frame(ws::WebSockets.WebSocket{Sockets.TCPSocket}, wsf::WebSockets.WebSocketFragment)
       @ WebSockets ~\.julia\packages\WebSockets\QcswW\src\WebSockets.jl:376
     [2] read(ws::WebSockets.WebSocket{Sockets.TCPSocket})
       @ WebSockets ~\.julia\packages\WebSockets\QcswW\src\WebSockets.jl:459
     [3] macro expansion
       @ ~\.julia\packages\WebIO\Fy9h1\src\providers\mux.jl:41 [inlined]
     [4] (::WebIO.var"#110#111"{WebIO.WebSockConnection, WebSockets.WebSocket{Sockets.TCPSocket}})()
       @ WebIO .\task.jl:406

using CSV, DataFrames, Interact, Plots
loadbutton = filepicker()
columnbuttons = Observable{Any}(dom"div"())
data = Observable{Any}(DataFrame)
plt = Observable{Any}(plot())
map!(CSV.read, data, loadbutton)

function makebuttons(df)
    buttons = button.(string.(names(df)))
    for (btn, name) in zip(buttons, names(df))
        map!(t -> histogram(df[name]), plt, btn)
    end
    dom"div"(hbox(buttons))
end

map!(makebuttons, columnbuttons, data)

ui = dom"div"(loadbutton, columnbuttons, plt)

Runge-Kutta 4-point method

function Runge_Kutta(init_conditions, parameters, functions)
    # Initial conditions
    S0 = init_conditions[1]
    I0 = init_conditions[2]
    R0 = init_conditions[3]
    
    # Parameters (beta, gamma, N, days, h)
    beta = parameters[1]
    gamma = parameters[2]
    N = parameters[3]
    days = parameters[4]
    h = parameters[5]
    
    # Unpack the ODE functions
    dS = functions[1]
    dI = functions[2]
    dR = functions[3]
    
    # Initialise empty result arrs
    S_res = Any[]
    I_res = Any[]
    R_res = Any[]

    # Fill result arrs with initial conditions
    push!(S_res, S0)
    push!(I_res, I0)
    push!(R_res, R0)
    
    # Perform iterations
    for i in 1:days
        k1 = dS(S_res[i], I_res[i], R_res[i], N, beta, gamma)
        l1 = dI(S_res[i], I_res[i], R_res[i], N, beta, gamma)
        m1 = dR(S_res[i], I_res[i], R_res[i], N, beta, gamma)

        k2 = dS(S_res[i] + k1*h/2, I_res[i] + l1*h/2, R_res[i] + m1*h/2, N, beta, gamma)
        l2 = dI(S_res[i] + k1*h/2, I_res[i] + l1*h/2, R_res[i] + m1*h/2, N, beta, gamma)
        m2 = dR(S_res[i] + k1*h/2, I_res[i] + l1*h/2, R_res[i] + m1*h/2, N, beta, gamma)

        k3 = dS(S_res[i] + k2*h/2, I_res[i] + l2*h/2, R_res[i] + m2*h/2, N, beta, gamma)
        l3 = dI(S_res[i] + k2*h/2, I_res[i] + l2*h/2, R_res[i] + m2*h/2, N, beta, gamma)
        m3 = dR(S_res[i] + k2*h/2, I_res[i] + l2*h/2, R_res[i] + m2*h/2, N, beta, gamma)

        k4 = dS(S_res[i] + k3*h, I_res[i] + l3*h, R_res[i] + m3*h, N, beta, gamma)
        l4 = dI(S_res[i] + k3*h, I_res[i] + l3*h, R_res[i] + m3*h, N, beta, gamma)
        m4 = dR(S_res[i] + k3*h, I_res[i] + l3*h, R_res[i] + m3*h, N, beta, gamma)

        S_next = S_res[i] + (h/6)*(k1 + 2*k2 + 2*k3 + k4)
        I_next = I_res[i] + (h/6)*(l1 + 2*l2 + 2*l3 + l4)
        R_next = R_res[i] + (h/6)*(m1 + 2*m2 + 2*m3 + m4)

        push!(S_res, S_next)
        push!(I_res, I_next)
        push!(R_res, R_next)
    end
    
    return S_res, I_res, R_res
end

Runge_Kutta (generic function with 3 methods)

Initial conditions

# Define model parameters
N = 1000.0
beta = 0.15
gamma = 1.0/14.0

# Initial conditions (populations)
I0 = 1;
R0 = 0;
S0 = N - I0 - R0;

initial_conditions = (S0, I0, R0)

# Define additional parameters
days = 200
h = 1

parameters = (beta, gamma, N, days, h)

# Define the ODE functions
dS(S, I, R, N, beta, gamma) = -beta * S * I / N
dI(S, I, R, N, beta, gamma) = beta * S * I / N - gamma * I
dR(S, I, R, N, beta, gamma) = gamma * I

funcs = (dS, dI, dR)

#Runge-Kutta
output = Runge_Kutta(initial_conditions, parameters, funcs)

(Any[999.0, 998.8441304369333, 998.6755766631111, 998.4933135009048, 998.2962341069097, 998.0831436688928, 997.8527526508146, 997.6036695595166, 997.334393206227, 997.0433044358205  …  180.34647571763455, 180.22810431106194, 180.11494567962217, 180.00676908500378, 179.90335410097836, 179.80449014441473, 179.70997602827376, 179.61961953550357, 179.53323701281147, 179.45065298334268], Any[1, 1.081565866807174, 1.1697562761490417, 1.2651044229837707, 1.3681855976208945, 1.4796203695148034, 1.6000779878909697, 1.7302800097444866, 1.8710041654523482, 2.023088471738235  …  4.474993142981863, 4.280711680820616, 4.094793838883087, 3.9168856822428113, 3.7466477256132227, 3.5837543897281514, 3.4278934739904905, 3.27876564530966, 3.1360839429993757, 2.999573299564527], Any[0, 0.07430369625960788, 0.15466706073990164, 0.2415820761114341, 0.3355802954695052, 0.43723596159245814, 0.547169361294542, 0.666050430739052, 0.7946026283207424, 0.933607092441423  …  815.1785311393836, 815.4911840081174, 815.7902604814948, 816.0763452327534, 816.3499981734084, 816.6117554658571, 816.8621304977357, 817.1016148191868, 817.3306790441892, 817.5497737170929])

length(R_r);
plt4 = plot([output[1] output[2] output[3]])

plot(plot2, plt4)